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FractalCity Tutorial

Edition: 1

Publucation year: 2023

Author

• Renan Lucas Fagundes, Universidade Federal de Lavras, UFLA, MG.

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Content

1 Imports

• 1.1 Versions

2 Read Data

3 Generating the Street Network

• 3.1 Visualizing the Street Network
• 3.2 $\mu$-number (cyclomatic)
• 3.3 $\alpha$-number
• 3.4 $\beta$-number
• 3.5 $\gamma$-number
• 3.6 Street Network Area (convex hull)
• 3.7 Street Network Perimeter (convex hull)

4 Applying the 'SandBox' Method

• 4.1 Street Network

  • 4.1.1 Vizualizing the Schema
  • 4.1.2 Plot the Data and Average
  • 4.1.3 Generalized Fractal Dimension
  • 4.1.4 Box-counting, Information and Correlation Fractal Dimensions
  • 4.1.5 Mass Exponents
  • 4.1.6 Multifractal Spectrum

• 4.2 Population

  • 4.2.1 Vizualizing the Schema
  • 4.2.2 Plot the Data and Average
  • 4.2.3 Generalized Fractal Dimension
  • 4.2.4 Box-counting, Information and Correlation Fractal Dimensions
  • 4.2.5 Mass Exponents
  • 4.2.6 Multifractal Spectrum

5 Chemical Distance

• 5.1 Efficiency Fractal Dimension
• 5.2 Average Straightness
• 5.3 Average Sinuosity

6 Street Network and Population Plot

References

1. Imports

import numpy as np
import math
import pandas as pd
import geopandas as gpd
import networkx as nx
import powerlaw
import shapely.ops as so
import shapely.wkt
import matplotlib.pyplot as plt

from shapely import ops
from shapely.geometry import Point, box, MultiPoint, LineString, Polygon, MultiPolygon, MultiLineString, GeometryCollection
from shapely.strtree import STRtree

from FractalCity import *

1.1 Versions

np.__version__, pd.__version__, gpd.__version__, nx.__version__, powerlaw.__version__, shapely.__version__

('1.19.5', '1.1.5', '0.7.0', '2.5.1', '1.5', '1.8.4')

2 Read Data

data = ReadData('/home/renan/Documentos/Análise de dados/')

df_nodes, df_links = data.read_nodes_and_links('Lavras', 'MG', 'Sudeste')

df_nodes.info()

<class 'pandas.core.frame.DataFrame'>
RangeIndex: 3510 entries, 0 to 3509
Data columns (total 6 columns):
 #   Column    Non-Null Count  Dtype  
---  ------    --------------  -----  
 0   osmid     3510 non-null   int64  
 1   y         3510 non-null   float64
 2   x         3510 non-null   float64
 3   lon       3510 non-null   float64
 4   lat       3510 non-null   float64
 5   geometry  3510 non-null   object 
dtypes: float64(4), int64(1), object(1)
memory usage: 150.9+ KB

df_nodes.head()

	osmid	y	x	lon	lat	geometry
0	353501750	7.650208e+06	499626.949178	-45.003596	-21.249785	POINT (-45.0035955 -21.2497847)
1	1431958418	7.650181e+06	499606.364850	-45.003794	-21.250029	POINT (-45.0037939 -21.2500289)
2	1432666855	7.650172e+06	499627.344245	-45.003592	-21.250102	POINT (-45.0035917 -21.2501023)
3	353502592	7.649695e+06	500488.805280	-44.995289	-21.254418	POINT (-44.9952887 -21.254418)
4	3901738964	7.649714e+06	500355.526301	-44.996573	-21.254240	POINT (-44.9965733 -21.2542403)

df_links.info()

<class 'pandas.core.frame.DataFrame'>
RangeIndex: 9187 entries, 0 to 9186
Data columns (total 4 columns):
 #   Column    Non-Null Count  Dtype  
---  ------    --------------  -----  
 0   u         9187 non-null   int64  
 1   v         9187 non-null   int64  
 2   length    9187 non-null   float64
 3   geometry  9187 non-null   object 
dtypes: float64(1), int64(2), object(1)
memory usage: 251.3+ KB

df_links.head()

	u	v	length	geometry
0	353501750	1431958418	36.585741	LINESTRING (-45.0035955 -21.2497847, -45.00379...
1	353501750	1432666855	37.936648	LINESTRING (-45.0035955 -21.2497847, -45.00359...
2	1431958418	1432666855	24.155855	LINESTRING (-45.0037939 -21.25002890000001, -4...
3	1431958418	1432666850	66.633354	LINESTRING (-45.0037939 -21.25002890000001, -4...
4	1432666855	1430927687	142.154239	LINESTRING (-45.0035917 -21.25010230000001, -4...

3 Generating the Street Network

sn = StreetNetwork()

#help(sn)

G = sn.generate_network(df_nodes, df_links)

3.1 Visualizing the Street Network

kwargs = {'figsize': (8,8),
          'node_size': 0,
          'edge_width': 1}

sn.plot_network(G, kwargs)

3.2 $\mu$-number (cyclomatic)

sn.cyclomatic_number(G)

1616

3.3 $\alpha$-number

sn.alpha_number(G)

0.23036350677120457

3.4 $\beta$-number

sn.beta_number(G)

1.46011396011396

3.5 $\gamma$-number

sn.gamma_number(G)

0.4869821360699354

3.6 Street Network Area (convex hull)

sn.network_area(G)/10e6

24.262611687709807

3.7 Street Network Perimeter (convex hull)

sn.network_perimeter(G)/10e3

5.969933874939586

4 Applying the 'SandBox' Method

sb = SandBox()

#help(sb)

4.1 Street Network

lon_red, lat_red, lon_blue, lat_blue, lon_gray, lat_gray = sb.generate_grid(df_nodes)

sb.generate_points(lon_red, lat_red, lon_blue, lat_blue, lon_gray, lat_gray)

Points created!

sb.apply_sandbox_on_street_network(num_div_rmin = 50, num_div_rmax = 2, num_intervals = 50, fraction = 4)

Sandbox method successfully applied!

street_network_data = sb.get_sandbox_data_from_street_network()

4.1.1 Vizualizing the Schema

lon_p, lat_p = [pos[0] for pos in street_network_data['analized_points']], [pos[1] for pos in street_network_data['analized_points']]

fig, ax = plt.subplots(figsize=(8,8))

ax.plot(lon_gray, lat_gray, 'o', markersize = 1, color='gray', label = 'grid')
ax.plot(lon_blue, lat_blue, 'o', markersize = 2, color='blue', label = 'street network nodes')
ax.plot(lon_red, lat_red, 'o', markersize = 3, color='red', label = 'points found')
ax.plot(lon_p, lat_p, 'o', markersize = 4, color='green', label = 'analized points')

ax.set_xlabel('lon')
ax.set_ylabel('lat')

plt.title('Lavras, MG')
plt.legend()

plt.show()

4.1.2 Plot the Data and Average

average = [np.mean([street_network_data['N'][i][j] for i in range(len(street_network_data['N']))]) for j in range(len(street_network_data['N'][0]))]

fig, ax = plt.subplots(figsize=(5,5))

ax.plot(street_network_data['r'], street_network_data['N'], 'o' , fillstyle = 'none')
ax.plot(street_network_data['r'][0], average, 'k-', label = 'average')

ax.set_xscale('log')
ax.set_yscale('log')

ax.set_xlabel(r'$r$')
ax.set_ylabel(r'$N(r)$')

plt.title('Street Network: Lavras, MG')
plt.legend()

plt.show()

4.1.3 Generalized Fractal Dimension

moment_list = list(np.linspace(-10, 10, 100))

dict_gen_dim = sb.street_network_generalized_dimension(street_network_data, moment_list)

fig, ax = plt.subplots(figsize=(5,5))

ax.plot(moment_list, dict_gen_dim['gen_dim_street_network'], 's', fillstyle='none', label = 'Generalized Fractal Dimension')

ax.set_xlabel(r'$q$')
ax.set_ylabel(r'$D_{q}$')

plt.title('Street Network: Lavras, MG')
plt.legend()

plt.show()

4.1.4 Box-counting, Information and Correlation Fractal Dimensions

q0_index = moment_list.index(0.10101010101010033)
q1_index = moment_list.index(1.1111111111111107)
q2_index = moment_list.index(2.121212121212121)

D0 = round(dict_gen_dim['gen_dim_street_network'][q0_index],2)
D0_std = round(dict_gen_dim['gen_dim_street_network_std'][q0_index], 2)
D0_r2 = round(dict_gen_dim['gen_dim_street_network_r2'][q0_index],2)


D1 = round(dict_gen_dim['gen_dim_street_network'][q1_index],2)
D1_std = round(dict_gen_dim['gen_dim_street_network_std'][q1_index], 2)
D1_r2 = round(dict_gen_dim['gen_dim_street_network_r2'][q1_index],2)

D2 = round(dict_gen_dim['gen_dim_street_network'][q2_index],2)
D2_std = round(dict_gen_dim['gen_dim_street_network_std'][q2_index], 2)
D2_r2 = round(dict_gen_dim['gen_dim_street_network_r2'][q2_index],2)

print(f'Box-counting: {D0,D0_std,D0_r2}\nInformation: {D1,D1_std,D1_r2}\nCorrelation: {D2,D2_std,D2_r2}', sep='\n')

Box-counting: (1.65, 0.03, 1.0)
Information: (1.58, 0.04, 0.99)
Correlation: (1.53, 0.06, 0.98)

4.1.4 Mass Exponents

tau_list = sb.street_network_mass_exponent(dict_gen_dim['gen_dim_street_network'], moment_list)

fig, ax = plt.subplots(figsize=(5,5))

ax.plot(moment_list, tau_list, 's', fillstyle='none', color='tab:green', label = 'Mass exponents')

ax.set_xlabel(r'$q$')
ax.set_ylabel(r'$\tau(q)$')

plt.title('Street Network: Lavras, MG')
plt.legend()

plt.show()

4.1.5 Multifractal Spectrum

alpha, falpha = sb.street_network_multifractal_spectrum(tau_list, moment_list)

fig, ax = plt.subplots(figsize=(5,5))

ax.plot(alpha, falpha, 's', fillstyle='none', color = 'tab:orange', label = 'Multifractal spectrum')

ax.set_xlabel(r'$\alpha$')
ax.set_ylabel(r'$f(\alpha)$')

plt.title('Street Network: Lavras, MG')
plt.legend()

plt.show()

4.2 Population

pol, popul = data.read_polygons_populations('Lavras', 'MG', 'Sudeste')

population_data = sb.apply_sandbox_on_population(pol, popul, lon_red, lat_red, lon_gray, lat_gray, num_div_rmin = 50, num_div_rmax = 2, num_intervals = 50, fraction = 4)

/home/renan/Documentos/Análise de dados/FractalCity.py:1058: ShapelyDeprecationWarning: STRtree will be changed in 2.0.0 and will not be compatible with versions < 2.
  tree = STRtree(multipolygon.geoms)

4.2.1 Vizualizing the schema

lon_p, lat_p = [pos[0] for pos in population_data['analized_points']], [pos[1] for pos in population_data['analized_points']]

fig, ax = plt.subplots(figsize=(8,8))

ax.plot(lon_gray, lat_gray, 'o', markersize = 1, color='gray', label = 'grid')
ax.plot(lon_blue, lat_blue, 'o', markersize = 2, color='blue', label = 'street network nodes')
ax.plot(lon_red, lat_red, 'o', markersize = 3, color='red', label = 'points found')
ax.plot(lon_p, lat_p, 'o', markersize = 4, color='green', label = 'analized points')

ax.set_xlabel('lon')
ax.set_ylabel('lat')

plt.title('Lavras, MG')
plt.legend()

plt.show()

4.2.2 Plot the data and the average

average = [np.mean([population_data['N'][i][j] for i in range(len(population_data['N']))]) for j in range(len(population_data['N'][0]))]

fig, ax = plt.subplots(figsize=(5,5))

ax.plot(population_data['r'], population_data['N'], 'o' , fillstyle = 'none')
ax.plot(population_data['r'][0], average, 'k-', label = 'average')

ax.set_xscale('log')
ax.set_yscale('log')

ax.set_xlabel(r'$r$')
ax.set_ylabel(r'$N(r)$')

plt.title('Population: Lavras, MG')
plt.legend()

plt.show()

4.2.3 Generalized Fractal Dimension

dict_gen_dim = sb.population_generalized_dimension(population_data, moment_list)

fig, ax = plt.subplots(figsize=(5,5))

ax.plot(moment_list, dict_gen_dim['gen_dim_population'], 'h', fillstyle='none', label = 'Generalized Dimension')

ax.set_xlabel(r'$q$')
ax.set_ylabel(r'$D_{q}$')

plt.title('Population: Lavras, MG')
plt.legend()

plt.show()

4.2.4 Box-counting, Information and Correlation Fractal Dimensions

q0_index = moment_list.index(0.10101010101010033)
q1_index = moment_list.index(1.1111111111111107)
q2_index = moment_list.index(2.121212121212121)

D0 = round(dict_gen_dim['gen_dim_population'][q0_index],2)
D0_std = round(dict_gen_dim['gen_dim_population_std'][q0_index], 2)
D0_r2 = round(dict_gen_dim['gen_dim_population_r2'][q0_index],2)


D1 = round(dict_gen_dim['gen_dim_population'][q1_index],2)
D1_std = round(dict_gen_dim['gen_dim_population_std'][q1_index], 2)
D1_r2 = round(dict_gen_dim['gen_dim_population_r2'][q1_index],2)

D2 = round(dict_gen_dim['gen_dim_population'][q2_index],2)
D2_std = round(dict_gen_dim['gen_dim_population_std'][q2_index], 2)
D2_r2 = round(dict_gen_dim['gen_dim_population_r2'][q2_index],2)

print(f'Box-counting: {D0,D0_std,D0_r2}\nInformation: {D1,D1_std,D1_r2}\nCorrelation: {D2,D2_std,D2_r2}', sep='\n')

Box-counting: (2.79, 0.1, 0.98)
Information: (1.95, 0.06, 0.99)
Correlation: (1.63, 0.05, 0.99)

4.2.5 Mass Exponents

tau_list = sb.population_mass_exponent(dict_gen_dim['gen_dim_population'], moment_list)

fig, ax = plt.subplots(figsize=(5,5))

ax.plot(moment_list, tau_list, 'h', fillstyle='none', color='tab:green', label = 'Mass exponents')

ax.set_xlabel(r'$q$')
ax.set_ylabel(r'$\tau(q)$')

plt.title('Population: Lavras, MG')
plt.legend()

plt.show()

4.2.6 Multifractal Spectrum

alpha, falpha = sb.population_multifractal_spectrum(tau_list, moment_list)

fig, ax = plt.subplots(figsize=(5,5))

ax.plot(alpha, falpha, 'h', fillstyle='none', color = 'tab:orange', label = 'Multifractal spectrum')

ax.set_xlabel(r'$\alpha$')
ax.set_ylabel(r'$f(\alpha)$')

plt.title('Population: Lavras, MG')
plt.legend()

plt.show()

5 Chemical Distance

dict_routes = sn.generate_routes(G)

euclidian_distance = dict_routes['euclidian_distance']
chemical_distance = dict_routes['chemical_distance']
average_euclidian_distance = dict_routes['average_euclidian_distance']
chemical_routes = dict_routes['chemical_routes']
chemical_routes_pos = dict_routes['chemical_routes_pos']

5.1 Efficiency Fractal Dimension

fig, ax = plt.subplots(figsize=(5,5))

ax.plot(average_euclidian_distance, chemical_distance, 'o', fillstyle='none', color = 'tab:blue', label = 'data')

ax.set_xscale('log')
ax.set_yscale('log')

ax.set_xlabel(r'$r$')
ax.set_ylabel(r'$\ell$')

plt.title('Chem. and Euclid. distance: Lavras, MG')
plt.legend()

plt.show()

5.2 Average Straightness

fig, ax = plt.subplots(figsize=(5,5))

average_straightness = np.divide(average_euclidian_distance, chemical_distance)

ax.plot(average_euclidian_distance, average_straightness, 'o', fillstyle='none', color = 'tab:blue', label = 'data')

ax.set_xscale('log')
#ax.set_yscale('log')

ax.set_xlabel(r'$r$')
ax.set_ylabel(r'$r/\ell$')

plt.title('Straightness: Lavras, MG')
plt.legend()

plt.show()

5.3 Average Sinuosity

fig, ax = plt.subplots(figsize=(5,5))

average_sinuosity = np.divide(chemical_distance, average_euclidian_distance)

ax.plot(average_euclidian_distance, average_sinuosity, 'o', fillstyle='none', color = 'tab:blue', label = 'data')

ax.set_xscale('log')
#ax.set_yscale('log')

ax.set_xlabel(r'$r$')
ax.set_ylabel(r'$\ell / r$')

plt.title('Sinuosity: Lavras, MG')
plt.legend()

plt.show()

6 Street Network and Population Plot

psp = PlotStreetPopulation(pol, popul, df_nodes, df_links)

#help(psp)

kwargs = {'legend': True,
          'markersize': 2,
          'linewidth': 1,
          'node_color': 'white',
          'edge_color': 'white',
          'figsize': (8, 8),
          'xlim': (-45.05, -44.95),
          'ylim': (-21.29, -21.20),
          'xlabel': 'lon',
          'ylabel': 'lat',
          'legend_kwds': {"label": "Population in 2010", "orientation": "vertical"},
          'cmap': 'viridis'}

psp.plot_geo_population_network(kwargs)

/home/renan/.local/lib/python3.6/site-packages/descartes/patch.py:65: ShapelyDeprecationWarning: The array interface is deprecated and will no longer work in Shapely 2.0. Convert the '.coords' to a numpy array instead.
  for t in polygon])
/home/renan/.local/lib/python3.6/site-packages/geopandas/plotting.py:183: ShapelyDeprecationWarning: The array interface is deprecated and will no longer work in Shapely 2.0. Convert the '.coords' to a numpy array instead.
  segments = [np.array(linestring)[:, :2] for linestring in geoms]

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